The Queen offered Ireland the nearest the royal family has ever come to an apology for Britain's actions in the tortured relations between the two countries, in a speech at a state banquet Dublin.

She told guests from the northern and southern Irish communities: "It is a sad and regrettable reality that through history our islands have experienced more than their fair share of heartache, turbulence and loss ... with the benefit of historical hindsight we can all see things which we wish had been done differently, or not at all."

The remarks, at a dinner in Dublin Castle, former headquarters of British rule in Ireland before independence in 1922, came as dissident republicans staged a small but violent demonstration.

Opening her speech in Irish with "A Úachtaráin agus a chairde [president and friends]", the Queen spoke of the importance of forbearance and conciliation, "of being able to bow to the past but not to be bound by it", and of the many who have suffered the painful legacy of loss. Lord Mountbatten, her husband's uncle, was killed by the IRA off the west coast of Ireland in August 1979. She said: "To all those who have suffered as a consequence of our troubled past I extend my sincere thoughts and deep sympathy."

The Queen spoke also of increasingly strong bonds and values: "The lessons of the peace process are clear: whatever life throws at us, our individual responsibilities will be all the stronger for working together and sharing the load ... The ties of family, friends and affection are our most precious resource ... the lifeblood of partnership across these islands, a golden thread runs through all our joint successes so far and all we will go on to achieve."

The Irish president, Mary McAleese, said the Queen's visit marked a new chapter in relations between the two countries "that may still be a work in progress, but happily has also become a work of progress, of partnership and friendship".

McAleese said: "I am particularly proud of this island's peacemakers who, having experienced first hand the appalling, toxic harvest of failing to resolve old hatreds and political differences, rejected the perennial culture of conflict and compromised enough to let a new future in."

Earlier, the Queen's reconciliation tour of the Irish Republic had taken in Croke Park, the home of Gaelic sport, in Dublin.

With its seating tiers, advert boards and video screens it seems the same as the many other sports grounds she has visited around the world: except that in a few minutes on Sunday 21 November 1920 British troops and Irish police fired into the crowd at a Gaelic football match between Tipperary and Dublin, killing 14 people, including one of the away team players, Michael Hogan. The stadium has, of course, changed utterly, out of all recognition, since then. It was one of the worst incidents of the savage Irish war of independence, the fact that it was a retaliation for the IRA's assassination of British undercover agents earlier in the day, no excuse and it has scarred Anglo-Irish relations for 90 years.

So the Queen's emergence from the players' tunnel in the Michael Hogan stand represented a second gesture of reconciliation in two days, after her wreath-laying on Tuesday at the city's garden of remembrance, honouring those killed in the fight for independence.

Christy Cooney, the Gaelic Athletic Association's president welcomed her, referring briefly to "tragic events" in the two countries' history and the loss of lives "including those who died in this place".

He called the visit "an important underpinning and advance of the process which … is now irreversible". Unspoken, but hanging heavy over proceedings, were thoughts of the young Northern Irish police officer, and Gaelic footballer, Ronan Kerr, murdered by the Real IRA in April: a previously unimaginable happening not least because members of the north's police were until recently banned from membership of the GAA.

Nevertheless, so tender are sensitivities that only one county GAA organisation from Northern Ireland attended the Queen's visit.

Outside the stadium, well out of earshot, 40 members of Republican Sinn Féin, aligned to the Continuity IRA, protested against the attempt to normalise relations with Britain. Later, outside Dublin Castle, missiles and fireworks were fired at Garda lines by up to 200 dissident republican demonstrators. The protesters, from three dissident Republican organisations — Republican Sinn Féin, the 32 County Sovereignty Movement and the socialist Irish republican party Éirígí, confronted the Garda to the side of the city's Christ Church cathedral, several hundred yards away from the back of Dublin Castle where the state banquet was taking place. The Garda made 20 arrests.

In the morning, the Queen and Duke of Edinburgh, accompanied by McAleese, had paid respects to more forgotten casualties of the last century: the 49,400 Irish who died fighting for Britain in the first world war. She laid a wreath at the memorial, officially dedicated only recently, to honour their memory.

Among those present was Jackie McDonald, a leader of the banned terrorist Ulster Defence Association; it was not clear whether the royal party realised he was present and there was no sign that the Queen recognised his presence.

Official representatives of Sinn Féin have been absent from this and other events, having asked not to be invited to meet the Queen, though Gerry Adams, the party's president, was seen briefly at a protest demonstration.

The Queen has been whisked through heavily policed streets nearly devoid of traffic and spectators. Those who turned up early outside the Guinness Storehouse visitor centre to see the royal couple only managed a distant glimpse of the car.

Eamonn Murphy, 66, a former brewery worker, was philosophical about the security. "History is history," he told reporters. "It is always there but it is like every disagreement or row that people have, there is always a way back. In this case it has taken a very long time, but the mere fact that she did what she did yesterday has gone a long way to bury the past."

In another sign of reconciliation, of a more mundane sort, On a more mundane level of reconciliation, Iris Robinson, wife of the Northern Ireland first minister, Peter Robinson, was due to attend the state dinner with her husband, her first public appearance since revelations last year of her affair with a teenager.

• This article was amended on 19 May 2011. The original said that only one county GAA organisation "from Ulster" attended the Queen's visit. It also said that the memorial honouring Ireland's first world war dead had been built only recently. This has been corrected.